number of real points

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European Journal of Mathematics ISSN 2199-675X

Volume 5 Number 3

European Journal of Mathematics (2019) 5:686-711

DOI 10.1007/s40879-019-00324-9

number of real points

Erwan Brugallé, Alex Degtyarev, Ilia

Itenberg & Frédéric Mangolte

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R E S E A R C H A R T I C L E

Real algebraic curves with large finite number of real points

Erwan Brugallé¹· Alex Degtyarev²· Ilia Itenberg³· Frédéric Mangolte⁴

Received: 10 July 2018 / Revised: 22 January 2019 / Accepted: 1 February 2019 / Published online: 1 March 2019

Abstract

We address the problem of the maximal finite number of real points of a real algebraic curve (of a given degree and, sometimes, genus) in the projective plane. We improve the known upper and lower bounds and construct close to optimal curves of small degree. Our upper bound is sharp if the genus is small as compared to the degree.

Some of the results are extended to other real algebraic surfaces, most notably ruled.

Keywords Positive polynomials· Real algebraic curves · Real algebraic surfaces · Patchworking

Mathematics Subject Classification 14P25· 14H50 · 14M25

The first author is partially supported by the Grant TROPICOUNT of Région Pays de la Loire. The first, third and fourth authors are partially supported by the ANR Grant ANR-18-CE40-0009 ENUMGEOM.

The second author is partially supported by the TÜB˙ITAK Grant 116F211.

B

Ilia Itenberg

ilia.itenberg@imj-prg.fr Erwan Brugallé

erwan.brugalle@math.cnrs.fr Alex Degtyarev

degt@fen.bilkent.edu.tr Frédéric Mangolte

frederic.mangolte@univ-angers.fr

1 Université de Nantes, Laboratoire de Mathématiques Jean Leray, 2 rue de la Houssinière, 44322 Nantes Cedex 3, France

2 Department of Mathematics, Bilkent University, 06800 Ankara, Turkey

3 Institut de Mathématiques de Jussieu–Paris Rive Gauche, Sorbonne Université, 4 place Jussieu, 75252 Paris Cedex 5, France

4 Laboratoire angevin de recherche en mathématiques (LAREMA), Université d’Angers, CNRS, 49045 Angers Cedex 01, France

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1 Introduction

A real algebraic variety(X, c) is a complex algebraic variety equipped with an anti- holomorphic involution c: X → X, called a real structure. We denote by RX the real part of X , i.e., the fixed point set of c. With a certain abuse of language, a real algebraic variety is called finite if so is its real part. Note that each real point of a finite real algebraic variety of positive dimension is in the singular locus of the variety.

1.1 Statement of the problem

In this paper we mainly deal with the first non-trivial case, namely, finite real algebraic curves inCP². (Some of the results are extended to more general surfaces.) The degree of such a curve C ⊂ CP²is necessarily even, deg C = 2k. Our primary concern is the number|RC| of real points of C.

Problem 1.1 For a given integer k 1, what is the maximal number δ(k) = max

|RC| : C ⊂ CP²a finite real algebraic curve, deg C = 2k

? For given integers k 1 and g 0, what is the maximal number

δg(k) = max

|RC| : C ⊂ CP²a finite real algebraic curve of genus g, deg C= 2k

? (See Sect.2for our convention for the genus of reducible curves.)

Remark 1.2 Since the curves considered are singular, we do not insist that they should be irreducible. The curves achieving the maximal possible value ofδ(k) are, indeed, reducible in degrees 2k= 2, 4 (see Sect.1.2), whereas they are irreducible in degrees 2k = 6, 8. It appears that in all degrees 2k 6 maximizing curves can be chosen irreducible.

The Petrovsky inequalities (see [16] and Remark2.3) result in the following upper bound:

|RC| 3

2k(k − 1) + 1.

Currently, this bound is the best known. Furthermore, being of topological nature, it is sharp in the realm of pseudo-holomorphic curves. Indeed, consider a rational simple Harnack curve of degree 2k inCP²(see [1,11,14]); this curve has(k − 1)(2k − 1) solitary real nodes (as usual, by a node we mean a non-degenerate double point, i.e., an A1-singularity) and an oval (see Remark2.3for the definition) surrounding (k −1)(k −2)/2 of them. One can erase all inner nodes, leaving the oval empty. Then, in the pseudo-holomorphic category, the oval can be contracted to an extra solitary node, giving rise to a finite real pseudo-holomorphic curve C ⊂ CP²of degree 2k with|RC| = 3k(k − 1)/2 + 1.

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1.2 Principal results

For the moment, the exact value ofδ(k) is known only for k 4. The upper (Petrovsky inequality) and lower bounds for a few small values of k are as follows:

k 1 2 3 4 5 6 7 8 9 10

δ(k) 1 4 10 19 31 46 64 85 109 136 δ(k) 1 4 10 19 30 45 59 78 98 123

The cases k = 1, 2 are obvious (union of two complex conjugate lines or conics, respectively). The lower bound for k = 6 is given by Proposition4.7, and all other cases are covered by Theorem4.5. Asymptotically, we have

4

3k² δ(k) 3 2k², where the lower bound follows from Theorem4.5.

A finite real sextic C6with|RC6| = δ(3) = 10 was constructed by David Hilbert [8]. We could not find in the literature a finite real octic C8with|RC8| = δ(4) = 19;

our construction given by Theorem4.5can easily be paraphrased without referring to patchworking. The best previously known asymptotic lower boundδ(k) 10k²/9 is found in Choi, Lam, Reznick [2].

With the genus g= g(C) fixed, the upper bound δg(k) k²+ g + 1

is also given by a strengthening of the Petrovsky inequalities (see Theorem2.5). In Theorem4.8, we show that this bound is sharp for g k − 3.

Most results extend to curves in ruled surfaces: upper bounds are given by The- orem 2.5(for g fixed) and Corollary 2.6; an asymptotic lower bound is given by Theorem4.2(which also covers arbitrary projective toric surfaces), and a few spo- radic constructions are discussed in Sects.5and6.

1.3 Contents of the paper

In Sect.2, we obtain the upper bounds, derived essentially from the Comessatti inequalities. In Sect.3, we discuss the auxiliary tools used in the constructions, namely, the patchworking techniques, hyperelliptic (aka bigonal) curves and dessins d’enfants, and deformation to the normal cone. Section4is dedicated to curves inCP²: we recast the upper bounds, describe a general construction for toric surfaces (Theorem4.2) and a slight improvement for the projective plane (Theorem4.5), and prove the sharpness of the boundδg(k) k²+ g + 1 for curves of small genus. In Sect.5, we consider surfaces ruled overR, proving the sharpness of the upper bounds for small bidegrees and for small genera. Finally, Sect.6deals with finite real curves in the ellipsoid.

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2 Strengthened Comessatti inequalities

Let(X, c) be a smooth real projective surface. We denote by σ_inv^±(X, c) (respectively, σ_skew^± (X, c)) the inertia indices of the invariant (respectively, skew-invariant) sublattice of the involution c_∗: H2(X; Z) → H2(X; Z) induced by c. The following statement is standard.

Proposition 2.1 (see, for example, [22]) One has σ_inv⁻(X, c) = 1

2(h¹^,1(X) + χ(RX)) − 1, σ_skew⁻ (X, c) = 1

2(h¹^,1(X) − χ(RX)), where h^•,•are the Hodge numbers andχ is the topological Euler characteristic.

Corollary 2.2 (Comessatti inequalities) One has

2− h¹^,1(X) χ(RX) h¹^,1(X).

Remark 2.3 Let C ⊂ CP²be a smooth real curve of degree 2k. Recall that an oval of C is a connected component o⊂ RC bounding a disk in RP²; the latter disk is called the interior of o. An oval o of C is called even (respectively, odd) if o is contained inside an even (respectively, odd) number of other ovals of C; the number of even (respectively, odd) ovals of a given curve C is denoted by p (respectively, n). The classical Petrovsky inequalities [16] state that

p− n 3

2k(k − 1) + 1, n− p 3

2k(k − 1).

These inequalities can be obtained by applying Corollary2.2to the double covering ofCP²branched along C ⊂ CP²(see, e.g., [22], [12, Theorem 3.3.14]).

The Comessatti and Petrovsky inequalities, strengthened in several ways (see, e.g., [21]), have a variety of applications. For example, for nodal finite real rational curves inCP²we immediately obtain the following statement.

Proposition 2.4 Let C ⊂ CP² be a nodal finite rational curve of degree 2k. Then,

|RC| k²+ 1.

Proof Denote by r the number of real nodes of C, and denote by s the number of pairs of complex conjugate nodes of C. We have r+ 2s = (k − 1)(2k − 1). Let, further, Y be the double covering ofCP²branched along the smooth real curve Ct ⊂ CP² obtained from C by a small perturbation creating an oval from each real node of C.

The union of r small discs bounded byRCt is denoted byRP₊²; let ¯c: Y → Y be the lift of the real structure such that the real part projects ontoRP₊². Each pair of complex conjugate nodes of C gives rise to a pair of ¯c∗-conjugate vanishing cycles in H2(Y ; Z); their difference is a skew-invariant class of square −4, and the s square −4 classes thus obtained are pairwise orthogonal.

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Since h¹^,1(Y ) = 3k²− 3k + 2 (see, e.g., [22]), Corollary2.2implies that χ(RY ) h¹^,1(Y ) − 2s = 3k(k − 1) + 2 − 2s = k²+ 1 + r.

Thus, r k²+ 1.

The above statement can be generalized to the case of not necessarily nodal curves of arbitrary genus in any smooth real projective surface.

Recall that the geometric genus g(C) of an irreducible and reduced algebraic curve C is the genus of its normalization. If C is reduced with irreducible components C1, . . . , Cn, the geometric genus of C is defined by

g(C) = g(C1) + · · · + g(Cn) + 1 − n.

In other words, 2− 2g(C) = χ(C), where C is the normalization.

Define also the weightp of a solitary point p of a real curve C as the minimal number of blow-ups at real points necessary to resolve p. More precisely,p = 1+

pi, the summation running over all real points piover p of the strict transform of C blown up at p. For example, the weight of a simple node equals 1, whereas the weight of an A2n−1-type point equals n. If|RC| < ∞, we define the weighted point countRC as the sum of the weights of all real points of C.

The topology of the ambient complex surface X is present in the next statement in the form of the coefficient

T2,1(X) =1 6

c²₁(X) − 5c2(X)

= 1

2(σ(X) − χ(X)) = b1(X) − h¹^,1(X) of the Todd genus (see [9]).

Theorem 2.5 Let(X, c) be a simply connected smooth real projective surface with non-empty connected real part. Let C ⊂ X be an ample reduced finite real algebraic curve such that[C] = 2e in H2(X; Z). Then, we have

|RC| RC e²+ g(C) − T2,1(X) + χ(RX) − 1. (1)

Furthermore, the second inequality is strict unless all singular points of C are double.

Proof Since [C] ∈ H2(X; Z) is divisible by 2, there exists a real double covering ρ : (Y , ¯c) → (X, c) ramified at C and such that ρ(RY ) = RX. By the embedded resolution of singularities, we can find a sequence of real blow-upsπi: Xi → Xi−1, i = 1, . . . , n, real curves Ci = π^∗Ci−1mod 2 ⊂ Xi, and real double coverings ρi = π_i^∗ρi−1: Yi → Xi ramified at Ci such that the curve Cn and surface Yn are nonsingular. (Here, a real blow-up is either a blow-up at a real point or a pair of blow- ups at two conjugate points. Byπ_i^∗Ci−1mod 2 we mean the reduced divisor obtained by retaining the odd multiplicity components of the divisorial pull-backπ_i^∗Ci−1.)

Using Proposition2.1, we can rewrite (1) in the form

e²+ g(C) + h¹^,1(X) − T2,1(X) + 2χ(RX) − 2σ_inv⁻(X, c) − RC 3.

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We proceed by induction and prove a modified version of the latter inequality, namely, e_i²+ g(Ci) + h¹^,1(Xi) − T2,1(Xi)

+ b⁻¹₁ (Yi) + 2χ(RXi) − 2σ_inv⁻(Xi, c) − RCi 3, (2)

where[Ci] = 2ei ∈ H2(Xi; Z) and b₁⁻¹(·) is the dimension of the (−1)-eigenspace ofρ∗on H1(·; C).

For the “complex” ingredients of (2), it suffices to consider a blow-upπ : X → X at a singular point p of C, not necessarily real, of multiplicity O 2. Denoting by C the strict transform of C, we have C = π^∗C mod 2= C+ εE, where E = π⁻¹(p) is the exceptional divisor and O= 2m + ε, m ∈ Z, ε = 0, 1. Then, in obvious notation,

e²= ˜e²+ m², g(C) = g(C) + ε, h¹^,1(X) = h¹^,1(X) − 1, T2,1(X) = T2,1(X) + 1.

Furthermore, from the isomorphisms H1(Y, ˜ρ^∗E) = H1(Y , p) = H1(Y ) we easily conclude that

b₁⁻¹(Y ) b⁻₁(Y) − b⁻₁( ˜ρ^∗E) b⁻¹₁ (Y) − 2(m − 1).

It follows that, when passing from X to X , the increment in the first five terms of (2) is at least(m − 1)²+ ε − 1 −1; this increment equals −1 if and only if p is a double point of C.

For the last three terms, assume first that the singular point p above is real. Then χ(RX) = χ(RX) + 1, σ_inv⁻(Xi, c) = σ_inv⁻(Xi, ˜c), RC = RC + 1, and the total increment in (2) is non-negative; it equals 0 if and only if p is a double point.

Now, letπ : X → X be a pair of blow-ups at two complex conjugate singular points of C. Then

χ(RX) = χ(RX), σ_inv⁻(Xi, c) = σ_inv⁻(Xi, ˜c) − 1, RC = RC, and, again, the total increment is non-negative, equal to 0 if and only if both points are double.

To establish (2) for the last, nonsingular, curve Cn, we use the following observa- tions:

• χ(Yn) = 2χ(Xn) − χ(Cn) (the Riemann–Hurwitz formula);

• σ (Yn) = 2σ(Xn) − 2e²n(Hirzebruch’s theorem);

• b1(Yn) − b1(Xn) = b⁻¹₁ (Yn), as b⁺¹₁ (Yn) = b1(Xn) via the transfer map;

• χ(RYn) = 2χ(RXn), since RCn= ∅ and RYn→ RXnis an unramified double covering.

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Then, (2) takes the form

σ_inv⁻(Yn, ¯cn) σ_inv⁻(Xn, cn),

which is obvious in view of the transfer map H2(Xn; R) → H2(Yn; R): this map is equivariant and isometric up to a factor of 2.

Thus, there remains to notice that b⁻¹₁ (Y0) = 0. Indeed, since C0= C is assumed ample, XC has homotopy type of a CW-complex of dimension 2 (as a Stein man- ifold). Hence, so does YC, and the homomorphism H1(C; R) → H1(Y ; R) is

surjective. Clearly, b⁻¹₁ (C) = 0.

Corollary 2.6 Let(X, c) and C ⊂ X be as in Theorem2.5. Then, we have

2|RC| 3e²− e ·c1(X) − T2,1(X) + χ(RX),

the inequality being strict unless each singular point of C is a solitary real node of RC.

Proof By the adjunction formula we have

g(C) 2e²− e ·c1(X) + 1 − |RC|,

and the result follows from Theorem2.5.

Remark 2.7 The assumptions π1(X) = 0 and b0(RX) = 1 in Theorem2.5are mainly used to assure the existence of a real double coveringρ : Y → X ramified over a given real divisor C. In general, one should speak about the divisibility by 2 of the real divisor class|C|R, i.e., class of real divisors modulo real linear equivalence. (If RX = ∅, one can alternatively speak about the set of real divisors in the linear system

|C| or a real point of Pic(X).) A necessary condition is the vanishing [C] = 0 ∈ H2n−2(X; Z/2Z), [RC] = 0 ∈ Hn−1(RX; Z/2Z),

where n= dim_C(X) and [RC] is the homology class of the real part of (any represen- tative of)|C| (the sufficiency of this condition in some special cases is discussed in Lemma3.4). If not empty, the set of double coverings ramified over C and admitting real structure is a torsor over the space of c^∗-invariant elements of H¹(X; Z/2Z).

The proof of the following theorem repeats literally that of Theorem2.5.

Theorem 2.8 Let(X, c) be a smooth real projective surface and C ⊂ X an ample finite reduced real algebraic curve such that the class|C|_Ris divisible by 2. A choice of a real double coveringρ : Y → X ramified over C defines a decomposition of RX into two disjoint subsetsRX₊= ρ(RY ) and RX₋consisting of whole components.

Then, we have

RC∩RX+ − RC∩RX− e²+ g(C) − T2,1(X) + χ(RX+) − χ(RX−) − 1, the inequality being strict unless all singular points of C are double.

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3 Construction tools 3.1 Patchworking

If is a convex lattice polygon contained in the non-negative quadrant (R0)²⊂ R², we denote by Tor() the toric variety associated with ; this variety is a surface if is non-degenerate. In the latter case, the complex torus(C^∗)²is naturally embedded in Tor(). Let V ⊂ (R₀)²∩ Z²be a finite set, and let P(x, y) =

(i, j)∈Vai jxⁱy^j be a real polynomial in two variables. The Newton polygonP of P is the convex hull inR²of those points in V that correspond to the non-zero monomials of P. The polynomial P defines an algebraic curve in the 2-dimensional complex torus(C^∗)²; the closure of this curve in Tor(P) is an algebraic curve C ⊂ Tor(P). If Q is a quadrant of(R^∗)²⊂ (C^∗)²and(a, b) is a vector in Z², we denote by Q(a, b) the quadrant

(x, y) ∈ (R^∗)²: ((−1)^ax, (−1)^by) ∈ Q .

If e is an integral segment whose direction is generated by a primitive integral vec- tor(a, b), we abbreviate Q(e^⊥)^..= Q(b, −a). A real algebraic curve C ⊂ Tor() is said to be ¹₄-finite (respectively, ¹₂-finite) if the intersection of the real part RC with the positive quadrant(R_>0)²(respectively, the union(R_>0)²∪ (R_>0)²(1, 0) is finite.

Given an algebraic curve C ⊂ Tor() and an edge e of , we put Te(C) ^..= C∩ D(e), where D(e) is the toric divisor corresponding to e.

Fix a subdivisionS = {1, . . . , N} of a convex polygon ⊂ (R₀)²such that there exists a piecewise-linear convex functionν : → R whose maximal linear- ity domains are precisely the non-degenerate lattice polygons1, . . . , N. Let ai j, (i, j) ∈ ∩ Z², be a collection of real numbers such that ai j = 0 whenever (i, j) is a vertex ofS. This gives rise to N real algebraic curves Ck, k = 1, . . . , N: each curve Ck ⊂ Tor(k) is defined by the polynomial

P(x, y) =

(i, j)∈k∩Z²

ai jxⁱy^j

with the Newton polygonk.

Commonly, we denote by Sing(C) the set of singular points of an algebraic curve C.

Assume that each curve Ckis nodal and Sing(Ck) is disjoint from the toric divisors of Tor(k) (but Ck can be tangent with arbitrary order of tangency to some toric divisors). For each inner edge e = i ∩ j of S, the toric divisors corresponding to e in Tor(i) and Tor(j) are naturally identified, as they both are Tor(e). The intersection points of Ci and Cj with these toric divisors are also identified, and, at each such point p ∈ Tor(e), the orders of intersection of Ci and Cj with Tor(e) automatically coincide; this common order is denoted by mult p and, if mult p> 1, the point p is called fat. Assume that mult p is even for each fat point p and that the local branches of Ci and Cj at each real fat point p are in the same quadrant Qp⊂ (R^∗)².

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Each edge E of  is a union of exterior edges e of S; denote the set of these edges by {E} and, given e ∈ {E}, let k(e) be the index such that e ⊂ k(e). The toric divisor D(E) ⊂ Tor() is a smooth real rational curve whose real part RD(E) is divided into two halves RD_±(E) by the intersections with other toric divisors of Tor(); we denote by RD₊(E) the half adjacent to the positive quadrant of (R^∗)². Similarly, the toric divisor D(e) ⊂ Tor(k(e)) is divided into RD_±(e).

Theorem 3.1 (Patchworking construction; essentially, [19, Theorem 2.4]) Under the assumptions above, there exists a family of real polynomials P^(t)(x, y), t ∈ R_>0, with the Newton polygon, such that, for sufficiently small t, the curve C^(t) ⊂ Tor() defined by P^(t)has the following properties:

• the curve C^(t)is nodal and Sing(C^(t)) is disjoint from the toric divisors;

• if all curves C1, . . . , CN are ¹₂-finite (respectively, ¹₄-finite), then so is C^(t);

• there is an injective map

: N k=1

Sing(Ck) → Sing(C^(t)),

such that the image of each real point is a real point of the same type (solitary/non- solitary) and in the same quadrant of(R^∗)², and the image of each imaginary point is imaginary;

• there is a partition

Sing(C^(t)) image of =

p

p,

p running over all fat points, so that|p| = 2m −1 if mult p = 2m. The points in pare imaginary if p is imaginary and real and solitary if p is real; in the latter case, m− 1 of these points lie in Qpand the others m points lie in Qp(e^⊥p), where p∈ Tor(ep);

• for each edge E of , there is a bijective map

E:

e∈{E}

Te(Ck(e)) → TE(C^(t))

preserving the intersection multiplicity and the position of points inRD_±(·) or D(·)RD(·).

Proof To deduce the statement from [19, Theorem 2.4], one can use [18, Lemma 5.4 (ii)] and the deformation patterns described in [10, Lemmas 3.10 and 3.11] (cf.

also the curves C_∗,0,0in Lemma3.2).

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n+b+q

RB0

RB∞

n+b−q−1

p∞

p0

Fig. 1 The curve C_n_,b,q

3.2 Bigonal curves via dessins d’enfants

We denote by n, n 0, the Hirzebruch surface of degree n, i.e., n= P

O_CP¹(n)⊕O_CP¹ .

Recall that 0= CP¹×CP¹and 1is the blow-up ofCP²at a point. The bundle projection induces a mapπ : n → CP¹, and we denote by F a fiber ofπ; it is isomorphic toCP¹. The images ofO_CP¹ andO_CP¹(n) are denoted by B0 and B_∞, respectively; these curves are sections ofπ. The group H²( n; C) = H¹^,1(X; C) is generated by the classes of B0and F , and we have

[B0]²= n, [B∞]²= −n, [F]²= 0, B_∞∼ B0− nF, c1( n) = 2[B0] + (2 − n)[F].

(If n > 0, the exceptional section B_∞is the only irreducible curve of negative self- intersection.) In other words, we have D ∼ aB0+ bF for each divisor D ⊂ n, and the pair(a, b) ∈ Z²is called the bidegree of D. The cone of effective divisors is generated by B_∞and F , and the cone of ample divisors is{aB0+ bF : a, b > 0}.

In this section, we equipCP¹ with the standard complex conjugation, and the surface nwith the real structure c induced by the standard complex conjugation on O_CP¹(n). Unless n = 0, this is the only real structure on nwith nonempty real part.

In particular c acts on H²( n; C) as − Id, and so σ_inv⁻(X, c) = 0. The real part of n

is a torus if n is even, and a Klein bottle if n is odd. In the former case, the complement R n (RB0∪ RB_∞) has two connected components, which we denote by R n,±. Lemma 3.2 Given integers n> 0, b 0, and 0 q n + b − 1, there exists a real algebraic rational curve C = Cn,b,qin 2nof bidegree(2, 2b) such that (see Fig.1):

(i) all singular points of C are 2n+ 2b − 1 solitary nodes; n + b + q of them lie in R 2n,+, and the other n+ b − q − 1 lie in R 2n,−;

(ii) the real partRC has a single extra oval o, which is contained in R 2n,−∪RB0∪ RB∞and does not contain any of the nodes in its interior;

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(iii) each intersection p_∞^..= o ∩ B∞and p0^..= o ∩ B0consists of a single point, the multiplicity being 2b and 4n+ 2b − 2q, respectively; the points p0and p_∞are on the same fiber F .

This curve can be perturbed to a curve Cn,b,q ⊂ 2n satisfying conditions (i) and (ii) and the following modified version of condition (iii):

(iii) the oval o intersects B∞and B0at, respectively, b and 2n+b−q simple tangency points.

Note that Cn,b,qintersects B0in q additional pairs of complex conjugate points.

Proof Up to elementary transformations of 2n(blowing up the point of intersection C∩ B_∞and blowing down the strict transforms of the corresponding fibers) we may assume that b= 0 and, hence, C is disjoint from B_∞. Then, C is given by P(x, y) = 0, where

P(x, y) = y²+ a1(x)y + a2(x), deg ai(x) = 2in. (3) (Strictly speaking, ai are sections of appropriate line bundles, but we pass to affine coordinates and regard ai as polynomials.) We will construct the curves using the techniques of dessins d’enfants, cf. [3,4,15]. Consider the rational function f : CP¹→ CP¹given by

f(x) = a²₁(x) − 4a2(x) a₁²(x) .

(This function differs from the j -invariant of the trigonal curve C + B0 by a few irrelevant factors.) The dessin of C is the graphD^..= f⁻¹(RP¹) decorated as shown in Fig.2. In addition to×-, ◦-, and •-vertices, it may also have monochrome vertices, which are the pull-backs of the real critical values of f other that 0, 1, or ∞. This graph is real, and we depict only its projection to the disk D^..= CP¹/x ∼ ¯x, showing the boundary∂ D by a wide grey curve: this boundary corresponds to the real parts RC ⊂ R 2n → RP¹. Assuming that a1, a2have no common roots, the real special vertices and edges ofD have the following geometric interpretation:

• a ×-vertex x0corresponds to a double root of the polynomial P(x0, y); the curve is tangent to a fiber if val x0= 2 and has a double point of type Ap−1, p= ¹₂val x0, otherwise;

• a ◦-vertex x0corresponds to an intersectionRC ∩ RB0of multiplicity ¹₂val x0;

• the real part RC is empty over each point of a solid edge and consists of two points over each point of any other edge;

• the points of RC over two ×-vertices x1, x2are in the same halfR 2n,±if and only if one has

val zi = 0 mod 8, the summation running over all •-vertices zi

in any of the two arcs of∂ D bounded by x1, x2.

0 1 ∞

Fig. 2 Decoration of a dessin

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Fig. 3 The dessin Dn,0,0and its modifications

(For the last item, observe that the valency of each•-vertex is 0 mod 4 and the sum of all valencies equals 2 deg f = 8n; hence, the sum in the statement is independent of the choice of the arc.)

Now, to construct the curves in the statements, we start with the dessin Dn,0,0shown in Fig.3, left: it has 2n•-vertices, 2n ◦-vertices, and (2n + 1) ×-vertices, two bivalent and 2n−1 four-valent, numbered consecutively along ∂ D. To obtain Dn,0,q, we replace q disjoint embraced fragments with copies of the fragment shown in Fig.3, right; by choosing the fragments replaced around even-numbered ×-vertices, we ensure that the solitary nodes would migrate fromR 2n,−toR 2n,+. Finally,Dn,0,qis obtained from Dn,0,qby contracting the dotted real segments connecting the real◦-vertices, so that the said vertices collide to a single(8n − 4q)-valent one. Each of these dessins D gives rise to a (not unique) equivariant topological branched covering f: S²→ CP¹ (cf. [3,4,15]), and the Riemann existence theorem gives us an analytic structure on the sphere S² making f a real rational function CP¹ → CP¹. There remains to take for a1a real polynomial with a simple zero at each (double) pole of f and let

a2^..=¹₄a₁²(1 − f ).

Generalizing, one can consider a geometrically ruled surface π : n(O)^..= P(O⊕OB) → B,

where B is a smooth compact real curve of genus g 1 and O is a line bundle, deg O = n 0. If O is also real, the surface n(O) acquires a real structure; the sections B0

and B_∞are also real and we can speak aboutRB0, RB∞. The real line bundleO is said to be even if the GL(1, R)-bundle RO over RB is trivial (cf. Remark2.7). In this case, the real partR n(O) is a disjoint union of tori, one torus Ti over each real componentRiB of B, and each complement T_i^◦ ^..= Ti (RB0∪ RB_∞) is made of two connected components (open annuli).

A smooth compact real curve B of genus g is called maximal if it has the maximal possible number of real connected components: b0(RB) = g + 1.

Lemma 3.3 Let n, g be two integers, n g − 1 0. Then there exists an even real line bundle O of degree deg O = 2n over a maximal real algebraic curve B

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of genusg, and a nodal real algebraic curve Cn(g) ⊂ 2n(O) realizing the class 2[B0] ∈ H2( 2n(O); Z) such that

(i) RCn(g) ∩ T1consists of 2n solitary nodes, all in the same connected component of T₁^◦;

(ii) RCn(g) ∩ T2 is a smooth connected curve, contained in a single connected component of T₂^◦except for n real points of simple tangency of Cnand B0; (iii) RCn(g)∩Ti, i 3, is a smooth connected curve, contained in a single connected

component of T₂^◦except for one real point of simple tangency of Cnand B0. Note that we can only assert the existence of a ruled surface 2n(O): the analytic structure on B and line bundleO are given by the construction and cannot be fixed in advance.

Proof We proceed as in the proof of Lemma3.2, with the “polynomials” ai sections ofO^⊗iin (3) and half-dessinDn(g)/cBin the surface D^..= B/cB, which, in the case of maximal B, is a disk with g holes; as above, we have∂ D = RB. The following technical requirements are necessary and sufficient for the existence of a topological ramified covering f: B → CP¹(see [3,4]) with B the orientable double of D:

• each region (connected component of DD) should admit an orientation inducing on the boundary the orientation inherited fromRP¹(the order onR), and

• each triangular region (i.e., one with a single vertex of each of the three special types×, ◦, and • in the boundary) should be a topological disk.

(For example, in the dessins Dn,0,qin Fig.3the orientations are given by a chessboard coloring and all regions are triangles.)

The curve Cn(g) as in the statement is obtained from the dessin Dn(g) constructed as follows. If g= 1, then Dn(1) is the dessin in the annulus shown in Fig.4, left (which is a slight modification of Dn,0,n−1in Fig.3): it has 2n real four-valent×-vertices, n inner four-valent•-vertices, and 2n ◦-vertices, n real four-valent and n inner bivalent.

(Recall that each inner vertex in D doubles in B, so that the total valency of the vertices of each kind sums up to 8n = 2 deg f , as expected.) This dessin is maximal in the sense that all its regions are triangles. To pass fromDn(1) to Dn(1 + q), q n,

Fig. 4 The dessinDn(1) and its modifications

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we replace small neighbourhoods of q inner◦-vertices with the fragments shown in Fig.4, right, creating q extra boundary components.

Each dessin Dn(g) satisfies the two conditions above and, thus, gives rise to a ramified covering f: B → CP¹. The analytic structure on B is given by the Riemann existence theorem, andO is the line bundle OB

₁

2P( f )

, where P( f ) is the divisor of poles of f . (All poles are even.) Then, the curve in question is given by “equation” (3), with the sections ai ∈ H⁰(B; O^⊗i) almost determined by their zeroes: Z(a1) =

1

2P( f ) and Z(a2) = Z(1 − f ). Further details of this construction (in the more

elaborate trigonal case) can be found in [3,4].

Next few lemmas deal with the real lifts of the curves constructed in Lemma3.3 under a ramified double covering of 2n(O). First, we discuss the existence of such coverings, cf. Remark2.7.

Lemma 3.4 Let n(O) be a real ruled surface over a real algebraic curve B such that RB = ∅, and let D be a real divisor on X. Then there exists a real divisor E on X such that|D|_R= 2|E|_Rif and only if[RD] = 0 ∈ H1(RX; Z/2Z).

Proof By [7, Proposition 2.3], we have

Pic( n(O)) ZB0⊕Pic(B),

and this isomorphism respects the action induced by the real structures. Let

|D| = m|B0| + |D0|.

Then m= [RD]◦[RF] mod 2, where F is the fiber of the ruling over a real point p ∈ RB, and D0 = D ◦ B∞, so that[RD0] = [RD]◦[RB∞]. There remains to observe that|D0|Ris divisible by 2 inRPic(B) if and only if [RD0] = 0 ∈ H0(B; Z/2Z). The

“only if” part is clear, and the “if” part follows from the fact that D0can be deformed,

through real divisors, to(deg D0)p.

Lemma 3.5 Let X ^..= n(O) be a real ruled surface over a real algebraic curve B such thatRB = ∅, and let C be a reduced real divisor on X such that [RC] = 0 ∈ H1(RX; Z/2Z). Then, for any surface S ⊂ RX such that ∂ S = RC, there exists a real double covering Y → X ramified over C such that RY projects onto S.

Proof Pick one covering Y0 → X, which exists by Lemma 3.4, and let S0be the projection ofRY0. We can assume that S0∩ T1= S ∩ T1for one of the components T1ofRX. Given another component Ti, consider a pathγi connecting a point in Ti

to one on T1, and let ˜γi = γi+ c_∗γi; in view of the obvious equivariant isomorphism H1(Y ; Z/2Z) H1(B; Z/2Z), these loops form a partial basis for the space of c_∗- invariant classes in H1(X; Z/2Z). Now, it suffices to twist Y0(cf. Remark2.7) by a cohomology class sending ˜γi to 0 or 1 if S∩ Ti coincides with S0∩ Ti or with the

closure of its complement, respectively.

Lemma 3.6 Let n, g be two integers, n g−1 0, and let B, O, and Cn(g) ⊂ 2n(O) be as in Lemma3.3. Then there exists a real double covering n(O) → 2n(O)

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ramified along B0∪ B∞and such that the pullback of Cn(g) is a finite real algebraic curve C_n(g) ⊂ n(O) with

|RCn(g)| = 5n − 1 + g.

Proof By Lemma3.5, there exists a real double covering n(O) → 2n(O) ramified along the curve B0∪ B_∞, such that the pullback in n(O) of the curve Cn(g) from Lemma3.3is a finite real algebraic curve C_n(g). Each node of Cn(g) gives rise to two solitary real nodes of C_n(g), and each tangency point of Cn(g) and RB0gives rise to

an extra solitary node of C_n(g).

3.3 Deformation to the normal cone

We briefly recall the deformation to normal cone construction in the setting we need here, and refer for example to [5] for more details. Given X a non-singular algebraic surface, and B ⊂ X a non-singular algebraic curve, we denote by NB/X the normal bundle of B in X , its projective completion by EB = P(NB/X⊕OB), and we define B_∞= EB NB/X. Note that if both X and B are real, then so are EBand B_∞.

LetX be the blow-up of X ×C along B ×{0}. The projection X ×C → C induces a flat projectionσ : X → C, and one has σ⁻¹(t) = X if t = 0, and σ⁻¹(0) = X ∪ EB. Furthermore, in this latter case X ∩ EB is the curve B in X , and the curve B_∞in EB. Note that if both X and B are real, and if we equipC with the standard complex conjugation, then the mapσ is a real map.

Let C0= CX∪ CB be an algebraic curve in X∪ EBsuch that:

• CX ⊂ X is nodal and intersects B transversely;

• CB ⊂ EB is nodal and intersects B_∞ transversely; let a = [CB]◦[F] in H2(EB; Z);

• CX∩ B = CB∩ B_∞= CX∩ CB.

In the following two propositions, we use [20, Theorem 2.8] to ensure the existence of a deformation Ctinσ⁻¹(t) within the linear system |CX+ aB| of the curve C0in some particular instances. We denote byP the set of nodes of C0 (X ∩ EB), and by IX (resp.IB) the sheaf of ideals ofP ∩ X (resp. P ∩ EB).

Proposition 3.7 In the notation above, suppose that X ⊂ CP³is a quadric ellipsoid, and that B is a real hyperplane section. If C0is a finite real algebraic curve, then there exists a finite real algebraic curve C1in X in the linear system|CX+ aB| such that

|RC1| = |RC0|.

Proof One has the following short exact sequence of sheaves:

0−→ O(CX)⊗IX −→ O(CX) −→ O_P∩X −→ 0.

(To shorten the notation, we abbreviateO(D) = OX(D) for a divisor D ⊂ X when the ambient variety X is understood.) Since H¹(X, O(CX)) = 0, one obtains the following exact sequence:

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0−→ H⁰(X, O(CX)⊗IX) −→ H⁰(X, O(CX))

−→ H⁰(P ∩ X, O_P∩X) −→ H¹(X, O(CX)⊗IX) −→ 0.

The surface CP¹×CP¹ is toric and it is a classical application of Riemann–Roch Theorem that H⁰(X, O(CX)⊗IX) has codimension |P ∩ X| in H⁰(X, O(CX)) (see for example [17, Lemma 8 and Corollary 2]). Since h⁰(P ∩ X, O_P∩X) = |P ∩ X|, we deduce that

H¹(X, O(CX)⊗IX) = 0.

The curve B is rational, and the surface EB is the surface 2. In particular, EB is a toric surface and B_∞is an irreducible component of its toric boundary. Hence we analogously obtain

H¹

EB, O(CB− B_∞)⊗IB

= 0.

Hence by [20, Theorem 3.1], the proposition is now a consequence of [20, Theorem

2.8].

Recall that H⁰(EB, O(CB)⊗IB) is the set of elements of H⁰(EB, O(CB)) vanishing onP ∩ EB.

Proposition 3.8 Suppose that X = CP², that B is a non-singular real cubic curve, and that CX = ∅. If CBis a finite real algebraic curve and if H⁰(EB, O(CB)⊗IB) is of codimension|P| in H⁰(EB, O(CB)), then there exists a finite real algebraic curve C1inCP²of degree 3a such that

|RC1| = |RCB|.

Proof Recall that EB is a ruled surface over B, i.e., is equipped with aCP¹-bundle π : EB → B. By [7, Lemma 2.4], we have

Hⁱ(EB, O(CB)) Hⁱ(B∞, π∗O(CB)), i ∈ {0, 1, 2}.

In particular the short exact sequence of sheaves

0−→ O(CB− B∞) −→ O(CB) −→ OB_∞ −→ 0 gives rise to the exact sequence

0−→ H⁰(EB, O(CB− B∞)) −→ H⁰(EB, O(CB)) −→ H⁰(B∞, OB_∞)

−→ H¹(EB, O(CB− B_∞)) −→ H¹(EB, O(CB))−→ H^ι¹ ¹(B_∞, OB_∞) −→ 0.

Furthermore, by [6, Proposition 3.1] we have H¹(EB, O(CB− B∞)) = 0, hence the mapι1is an isomorphism.

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On the other hand, the short exact sequence of sheaves 0−→ O(CB)⊗IB−→ O(CB) −→ O_P−→ 0 gives rise to the exact sequence

0−→ H⁰(EB, O(CB)⊗IB) −→ H⁰(EB, O(CB))

r₁

−→ H⁰(P, O_P) −→ H¹(EB, O(CB)⊗IB)−→ H^ι² ¹(EB, O(CB)) −→ 0.

By assumption, the map r1is surjective, so we deduce that the mapι2is an isomorphism.

We denote by L0 the invertible sheaf on the disjoint union of EB andCP² and restricting toO(CB) and O_CP²on EBandCP²respectively. Finally, we denote byL0

the invertible sheaf onσ⁻¹(0) for which C0is the zero set of a section. The natural short exact sequence

0−→ L0⊗IB −→ L0⊗IB−→ OB −→ 0 gives rise to the long exact sequence

0−→ H⁰(σ⁻¹(0), L0⊗IB) −→ H⁰(EB, O(CB)⊗IB)⊕ H⁰(CP², O_CP²)

r₂

−→ H⁰(B, OB) −→ H¹(σ⁻¹(0), L0⊗IB) −→ H¹(EB, O(CB)⊗IB)

−→ Hι ¹(B, OB) −→ H²(σ⁻¹(0), L0⊗IB) −→ 0.

The restriction of the map r2to the second factor H⁰(CP², O_CP²) is clearly an iso- morphism, hence we obtain the exact sequence

0−→ H¹(σ⁻¹(0), L0⊗IB) −→ H¹(EB, O(CB)⊗IB)

−→ Hι ¹(B, OB) −→ H²(σ⁻¹(0), L0⊗IB) −→ 0.

Sinceι = ι1◦ ι2is an isomorphism, we deduce that H¹(σ⁻¹(0), L0⊗IB) = 0. Now

the proposition follows from [20, Theorem 2.8].

4 Finite curves inCP²

In the case X = CP², Theorem2.5and Corollary2.6specialize as follows.

Theorem 4.1 Let C ⊂ CP²be a finite real algebraic curve of degree 2k. Then,

|RC| k²+ g(C) + 1, (4)

|RC| 3

2k(k − 1) + 1.

In the rest of this section, we discuss the sharpness of these bounds.

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Fig. 5 Tilling ofR²

4.1 Asymptotic constructions

The following asymptotic lower bound holds for any projective toric surface with the standard real structure.

Theorem 4.2 Let ⊂ R²be a convex lattice polygon, and let Xbe the associated toric surface. Then, there exists a sequence of finite real algebraic curves Ck ⊂ X with the Newton polygon(Ck) = 2k, such that

klim→∞

1

k²|RCk| = 4

3Area(), where Area() is the lattice area of .

Remark 4.3 In the settings of Theorem 4.2, assuming X smooth, the asymptotic upper bound for finite real algebraic curves C ⊂ Xwith(C) = 2k is given by Theorem2.5:

|RC| 3

2Area().

Proof of Theorem4.2 There exists a (unique) real rational cubic C ⊂ (C^∗)²such that

• (C) is the triangle with the vertices (0, 0), (2, 1), and (1, 2);

• the coefficient of the defining polynomial f of C at each corner of (C) equals 1;

• RC ∩ R²_>0is a single solitary node.

Figure5shows a tilling ofR²by lattice congruent copies of(C). Intersecting this tilling with k and making an appropriate adjustment in the vicinity of the boundary, we obtain a convex subdivision of k containing¹₃k²Area()+O(k) copies of (C).

Now, to each of these copies, we associate the curve given by an appropriate monomial multiple of either f(x, y) or f (1/x, 1/y). Applying Theorem3.1, we obtain a real polynomial fk whose zero locus in R²_>0consists of ¹₃k²Area() + O(k) solitary nodes. There remains to let Ck = { fk(x², y²) = 0}.

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(a)k = 3l (b) k = 3l + 1 (c)k = 3l + 2 Fig. 6 Subdivision of k

Corollary 4.4 There exists a sequence of finite real algebraic curves Ck ⊂ CP², deg Ck = 2k, such that

k→+∞lim 1

k²|RCk| = 4 3.

In the next theorem, we tweak the “adjustment in the vicinity of the boundary” in the proof of Theorem4.2in the case X= CP².

Theorem 4.5 For any integer k 3, there exists a finite real algebraic curve C ⊂ CP² of degree 2k such that

|RC| =

⎧⎪

⎨

⎪⎩

12l²− 4l + 2 if k = 3l, 12l²+ 4l + 3 if k = 3l + 1, 12l²+ 12l + 6 if k = 3l + 2.

Proof Following the proof of Theorem4.2, we use the subdivision of the triangle k (with the vertices(0, 0), (k, 0), and (0, k)) shown in Fig.6. In the t-axis (t = x or y), the missing coefficients are chosen so that the truncation of the resulting polynomial to each segment of length 1, 2 or 3 is an appropriate monomial multiple of 1, (t − 1)² or(t − 1)²(t + 1), respectively. Thus, each segment of length 2 or 3 gives rise to a point of tangency of the t-axis and the curve{ fk = 0}, resulting in two extra solitary nodes of Ck. Similarly, each vertex of k contained in a segment of length 1 gives

rise to an extra solitary node of Ck.

Remark 4.6 The construction of Theorem4.5for k = 3, 4 can easily be performed without using the patchworking technique.

4.2 A curve of degree 12

The construction given by Theorem4.5is the best known if k 5. If k = 6, it can be improved from 43 to 45.